Superquadrics and Angle-Preserving Transformations

Over the past 20 years, a great deal of interest has developed in the use of computer graphics and numerical methods for three-dimensional design. Significant progress in geometric modeling is being made, predominantly for objects best represented by lists of edges, faces, and vertices. One long-term goal of this work is a unified mathematical formalism, to form the basis of an interactive and intuitive design environment in which designers can simulate three-dimensional scenes with shading and texture, produce usable design images, verify numerical machining-control commands, and set up finite-element meshwork for structural and dynamic analysis. A new collection of smooth parametric objects and a new set of three-dimensional parametric modifiers show potential for helping to achieve this goal. The superquadric primitives and angle-preserving transformations extend the traditional geometric primitives-quadric surfaces and parametric patches-used in existing design packages, producing a new spectrum of flexible forms. Their chief advantage is that they allow complex solids and surfaces to be constructed and altered easily from a few interactive parameters

Interactive visualization tools for topological exploration

Thesis (Ph.D.) - Indiana University, Computer Science, 1992This thesis concerns using computer graphics methods to visualize mathematical objects. Abstract mathematical concepts are extremely difficult to visualize, particularly when higher dimensions are involved; I therefore concentrate on subject areas such as the topology and geometry of four dimensions which provide a very challenging domain for visualization techniques. In the first stage of this research, I applied existing three-dimensional computer graphics techniques to visualize projected four-dimensional mathematical objects in an interactive manner. I carried out experiments with direct object manipulation and constraint-based interaction and implemented tools for visualizing mathematical transformations. As an application, I applied these techniques to visualizing the conjecture known as Fermat's Last Theorem. Four-dimensional objects would best be perceived through four-dimensional eyes. Even though we do not have four-dimensional eyes, we can use computer graphics techniques to simulate the effect of a virtual four-dimensional camera viewing a scene where four-dimensional objects are being illuminated by four-dimensional light sources. I extended standard three-dimensional lighting and shading methods to work in the fourth dimension. This involved replacing the standard "z-buffer" algorithm by a "w-buffer" algorithm for handling occlusion, and replacing the standard "scan-line" conversion method by a new "scan-plane" conversion method. Furthermore, I implemented a new "thickening" technique that made it possible to illuminate surfaces correctly in four dimensions. Our new techniques generate smoothly shaded, highlighted view-volume images of mathematical objects as they would appear from a four-dimensional viewpoint. These images reveal fascinating structures of mathematical objects that could not be seen with standard 3D computer graphics techniques. As applications, we generated still images and animation sequences for mathematical objects such as the Steiner surface, the four-dimensional torus, and a knotted 2-sphere. The images of surfaces embedded in 4D that have been generated using our methods are unique in the history of mathematical visualization. Finally, I adapted these techniques to visualize volumetric data (3D scalar fields) generated by other scientific applications. Compared to other volume visualization techniques, this method provides a new approach that researchers can use to look at and manipulate certain classes of volume data

Cartography of irregularly shaped satellites

Irregularly shaped satellites, such as Phobos and Amalthea, do not lend themselves to mapping by conventional methods because mathematical projections of their surfaces fail to convey an accurate visual impression of the landforms, and because large and irregular scale changes make their features difficult to measure on maps. A digital mapping technique has therefore been developed by which maps are compiled from digital topographic and spacecraft image files. The digital file is geometrically transformed as desired for human viewing, either on video screens or on hard copy. Digital files of this kind consist of digital images superimposed on another digital file representing the three-dimensional form of a body

New Method for Estimating Fractal Dimension in 3D Space and Its Application to Complex Surfaces

The concept of “surface modeling” generally describes the process of representing a physical or artificial surface by a geometric model, namely a mathematical expression. Among the existing techniques applied for the characterization of a surface, terrain modeling relates to the representation of the natural surface of the Earth. Cartographic terrain or relief models as three-dimensional representations of a part of the Earth's surface convey an immediate and direct impression of a landscape and are much easier to understand than two-dimensional models. This paper addresses a major problem in complex surface modeling and evaluation consisting in the characterization of their topography and comparison among different textures, which can be relevant in different areas of research. A new algorithm is presented that allows calculating the fractal dimension of images of complex surfaces. The method is used to characterize different surfaces and compare their characteristics. The proposed new mathematical method computes the fractal dimension of the 3D space with the average space component of Hurst exponent H, while the estimated fractal dimension is used to evaluate, compare and characterize complex surfaces that are relevant in different areas of research. Various surfaces with both methods were analyzed and the results were compared. The study confirms that with known coordinates of a surface, it is possible to describe its complex structure. The estimated fractal dimension is proved to be an ideal tool for measuring the complexity of the various surfaces considered

New method for estimating fractal dimension in 3d space and its application to complex surfaces

The concept of “surface modeling” generally describes the process of representing a physical or artificial surface by a geometric model, namely a mathematical expression. Among the existing techniques applied for the characterization of a surface, terrain modeling relates to the representation of the natural surface of the Earth. Cartographic terrain or relief models as threedimensional representations of a part of the Earth's surface convey an immediate and direct impression of a landscape and are much easier to understand than two-dimensional models. This paper addresses a major problem in complex surface modeling and evaluation consisting in the characterization of their topography and comparison among different textures, which can be relevant in different areas of research. A new algorithm is presented that allows calculating the fractal dimension of images of complex surfaces. The method is used to characterize different surfaces and compare their characteristics. The proposed new mathematical method computes the fractal dimension of the 3D space with the average space component of Hurst exponent H, while the estimated fractal dimension is used to evaluate, compare and characterize complex surfaces that are relevant in different areas of research. Various surfaces with both methods were analyzed and the results were compared. The study confirms that with known coordinates of a surface, it is possible to describe its complex structure. The estimated fractal dimension is proved to be an ideal tool for measuring the complexity of the various surfaces considered

Entrepreneurial Activity for Integrating Mathematics, Art, and Social Science into Society

This article shows how entrepreneurial activity is used in a mathematical research product. The used methods are online and offline marketing activities where mathematical formulas are implemented to create visual images, both 2-dimensional and 3-dimensional, namely parametric curves mapped with complex functions and using algebraic surfaces. The results of these activities are motifs built into the form of souvenirs, ornaments, accessories, and batik motifs. Furthermore, entrepreneurial activities were carried out with students, and small business craftsmen collaborated. The main result of this activity is the integration of mathematics with social and humanities activities. This has implications for the development of science and collaboration in economic recovery. Additionally, this activity provides production opportunities and provides new insights for ordinary people that mathematics can be expressed in batik motifs, accessories, souvenirs, and various derivative products. Keywords: entrepreneur; mathematics; parametric; algebraic surface

Finite element surface registration incorporating curvature, volume preservation, and statistical model information

We present a novel method for nonrigid registration of 3D surfaces and images. The method can be used to register surfaces by means of their distance images, or to register medical images directly. It is formulated as a minimization problem of a sum of several terms representing the desired properties of a registration result: smoothness, volume preservation, matching of the surface, its curvature, and possible other feature images, as well as consistency with previous registration results of similar objects, represented by a statistical deformation model. While most of these concepts are already known, we present a coherent continuous formulation of these constraints, including the statistical deformation model. This continuous formulation renders the registration method independent of its discretization. The finite element discretization we present is, while independent of the registration functional, the second main contribution of this paper. The local discontinuous Galerkin method has not previously been used in image registration, and it provides an efficient and general framework to discretize each of the terms of our functional. Computational efficiency and modest memory consumption are achieved thanks to parallelization and locally adaptive mesh refinement. This allows for the first time the use of otherwise prohibitively large 3D statistical deformation models